#7094681

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,6,4,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,27)(22,24)(23,28)(25,30)(31,43)(32,39)(33,38)(34,42)(35,41)(36,37)(40,44)(45,47)(46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,33)(21,22)(25,39)(27,28)(29,37)(31,45)(34,35)(38,42)(40,43)(44,46);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,31)(21,36)(22,32)(23,30)(24,37)(26,40)(27,39)(28,29)(33,46)(34,45)(35,44)(38,47)(41,43)(42,48);(11,16)(13,15)(17,20,23,35)(18,26,28,41)(19,31,30,44)(21,34,22,33)(24,38,27,42)(25,40,29,43)(32,46,36,45)(37,47,39,48);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,32,30,36)(20,34,35,33)(25,39,29,37)(26,38,41,42)(31,46,44,45)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7001877);